New Level-0 Action of $U_q(widehat{sl}_2)$ on Leve

arXiv:q-alg/9505009v426Sep1995RIMS-1016q-alg/9505009May1995revised26September1995NewLevel-0ActionofUq(bsl2)onLevel-1Modules∗MichioJimboDepartmentofMathematics,FacultyofScience,KyotoUniversity,Kyoto606,JapanE-mail:[e-mail]ituteforMathematicalSciences,KyotoUniversity,Kyoto606,JapanE-mail:[e-mail]ituteforTheoreticalPhysics,KyotoUniversity,Kyoto606,JapanE-mail:[e-mail]tituteforMathematicalSciences,KyotoUniversity,Kyoto606,JapanE-mail:[e-mail]eforMathematicalSciences,KyotoUniversity,Kyoto606,JapanE-mail:[e-mail]finealevel-0actionofUq(bsl2)onthesumoflevel-1irreduciblehighestweightmodules.WiththeaidoftheaffineHeckealgebras,thisactionisrealizedonthebasiscreatedbythevertexoperators.Thisisaq-analogueoftheYangiansymmetryinconformalfieldtheory.1.IntroductionAttheUSCmeeting,anewsymmetrystructureinconformalfieldtheory(CFT),theYangiansymmetryofthespinonbasis,wasintroduced7.Theaimofthispaperistoinvestigatetheq-deformationofthismathematicalstructure.Beforestatingthemainachievementinthispaper,letusbrieflyreviewthedevelopmentsinthissubject.ResultssuggestingapossiblefermionicdescriptionoftheCFTspectrumwere∗contributiontotheProceedingsofthemeeting‘StatisticalMechanicsandQuantumFieldTheory’,attheUniversityofSouthernCalifornia,May16-21,1994firstobtainedbyRef.26forcertaincriticallatticemodels.Closelyrelatedresultsappearedearlierinthemathematicalliterature(seee.g.,Ref.28).AnalysisofthespectrumofthesemodelsfromtheBetheansatzyieldedfermionicexpressionsfortheconformalpartitionfunctions(seeRefs.25,30,27,29,18,1,2,34,20forfurtherdevelopments).ThissuggestedapossiblealternativedescriptionofthestatesinCFT,thespinonbasisofRefs.4,6.Theunderlyingalgebraicstructure,theYangiansymmetry,hasbeensuggestedfromquiteadifferentdirection.TherelevantsubjectwasthediagonalizationoftheCalogero–Sutherlandmodelwithspindegreesoffreedom8,33,3andtheHaldane–Shastrymodel22,32.TheirspectrumisdescribedbyanasymptoticBetheansatz33.Howeverthemultiplicityofstateshadneverbeenexplaineduntiltheworks23,3,whichshowthatthemodelhasasymmetrygeneratedbyaYangian13.ThemultiplicitiesareidentifiedwiththedimensionsofthehighestweightrepresentationsoftheYangian15,9.Itturnsoutthatthelevel-1integrablehighestweightmodule(IHWM)oftheaffineLiealgebrabsl2hasthesameYangianstructureasthespaceofstatesoftheselongrangeinteractionmodelsataspecialvalueofthecouplingconstant.Thisfactwasestablishedbyconstructingthenewspinonbasisinthelevel-1IHWM4,6andconsideringtheactionoftheYangianonit.Thespinonbasisisgeneratedbythespin-1/2primaryfieldoperatorinthelevel-1su(2)WZWmodel.Thecharacterformuladerivedinthespinonbasiscoincideswiththefermionicrepresentation.Recently,theextensionofspinonbasistothehigherlevelcasehasalsobeeninvestigated5.Nowletusturntoourmotivationinthiswork.InRefs.16,17,FaddeevandTakhta-jananalysedthestructureofthespaceofthestatesintheXXXmodelintheinfinitevolumelimit.InRefs.12,24,thesymmetrystructureoftheoff-criticalXXZmodelintheinfinitevolumelimitwasdeterminedbymeansoftherepresentationtheoryofthequantumaffinealgebraUq(bsl2).Thestructureofthespaceofstateswasunderstoodasthelevel-0representationonEndC(H)whereHisthedirectsumofthelevel-1irreduciblehighestweightUq(bsl2)-modules.Thevacuumstatesareidentifiedwithcertaindegreeoperators,andtheexcitedstatesarecreatedbythetypeIIvertexoperatoractingontheformer.ThisissimilartotheYangianstructurewementionedabove,inthesensethatwehavealevel-0action.However,thespaceEndC(H)ismuchbiggerthanthespaceHitself,whichgoestotheIHWMintheconformallimit.Itisstillunclearhowthelevel-0structureonEndC(H)isrelatedtotheYangianstructureonHinconformalfieldtheory.Asforthispoint,seetherecentpaperbyNakayashikiandYamada31.ThemathematicalcontentoftheYangianstructurewasknownsinceDrinfeldandothers14,11.Namely,thereisafunctorialconstructionofYangianrepresentationsfromcertainrepresentationsofthedegenerateaffineHeckealgebra.InRef.10,ChariandPressleygaveaconstructionoflevel-0Uq(bsln)-modulesfromfinite-dimensionalrepresentationsoftheaffineHeckealgebra.Inthispaper,weapplytheirconstructiontothelevel-1Uq(bsl2)-moduleHandobtainanewlevel-0actiononit.Forthispurpose,followingRef.6,weusethebasisofHcreatedbythetypeIvertexoperator.2TheconstructionofsuchabasiswasdiscussedinRef.19inamoregeneralcontextcorrespondingtotheXYZmodel.Wedonotknowifthe‘new’level-0actionofUq(bsl2)onlevel-1modulesisrelatedtosomephysicalmodels,inawaytheYangianstructureis.Beforeclosingtheintroductionletuspointoutsometechnicaldetails,exceptforwhichourconstructionisastraightforwardgeneralizationofthemethodsgiveninRefs.10,4,3,6.(i)Weareforcedtoconsiderinfinitesumsofproductsofthecomponentsofvertexoperators.Wewillintroducesuitablecompletionofthe‘N-spinon’spacetoclarifythemathematicalcontent.(ii)Becauseofthefusionrelation,whichconnectsthe‘N-spinon’sectortothe‘(N−2)-spinon’sector,wemustcheckthecompatibilityofthelevel-0actioninthesedifferentsectors.Thisistechnicallyno